An acoustic field has two physical characteristics that can be sensed: pressure and velocity. Pressure is a scalar quantity whereas velocity is a vector quantity. Conventional studio microphones sense one of these quantities or a linear combination of the two. An ‘omnidirectional’ microphone senses pressure, while a ‘figure-of-eight’ microphone senses velocity (or ‘pressure gradient’, which is closely related to velocity). Other types (subcardioid, cardioid, supercardioid and hypercardioid) sense a linear combination of pressure and velocity.
A way to express the far-field directional behaviour of a microphone is to expand its angular response into spherical harmonics. This expansion is the spherical equivalent of the more familiar Fourier series expansion of a periodic function of a single variable. Using the notation of Furze and Malham (described in Malham, D., “Second and Third Order Ambisonics—the Furse-Malham Set” http://www.york.ac.uk/inst/mustech/3d_audio/secondor.html) there is a single spherical harmonic of order 0 (zero) denoted by ‘W’, there are three harmonics of order 1 (one) denoted by ‘X’, ‘Y’ and ‘Z’, five of order 2 (two) denoted by ‘R’, ‘S’, ‘T’, ‘U’ and ‘V’, and so on. Pictures of these harmonics may be found in Leese, M. J., “Spherical Harmonic Components” at http://members.tripod.com/martin_leese/Ambisonic/harmonic.html.
The ideal omnidirectional microphone has a response independent of angle and is thus proportional to the zeroth-order harmonic W. The ideal figure-of-eight microphone has a response that is given by a linear combination of the three first-order harmonics X, Y and Z. The coefficients of the combination depend on the orientation of the microphone. Microphones of type ‘cardioid’ and its variants have a response that is a combination of W, X, Y and Z. All normal studio microphones are classified as ‘first order’ because their responses are linear combinations of harmonics of order 0 and 1.
If a microphone directivity could be synthesised using second order or higher order components also, then the directional resolution could be increased substantially. However there is no known physical quantity that is associated directly with a second or higher order spherical harmonic. Accordingly, higher-order responses have usually been synthesised using collections of slightly spaced microphone sensors or ‘capsules’, the outputs from which are processed to synthesise the desired directional response or responses. An early example of this technique is due to Blumlein, A. D. in “Improvements in and relating to Electrical Sound Transmission Systems”, British patent 456,444 (1936).
Various geometrical arrangements of microphone capsules are possible, but recently there has been considerable interest in capsules placed on the surface of a sphere. The sphere may exist physically, or merely be conceptual.
In British patent GB1512514 (“Coincident microphone simulation covering three dimensional space and yielding various directional outputs” 1977, filed July 1974), Craven, P. G. and Gerzon, M. A. disclose that the capsules may be placed at the points of a suitable integration rule for the sphere, and an output with spherical harmonic directivity can be obtained by multiplying each capsule output firstly by the value of the spherical harmonic at the capsule's position, and secondly by an integration weight given by the integration rule. This procedure assumes that each capsule is omnidirectional or, if it has directivity (for example cardioid), its direction of maximum sensitivity is pointed radially outward from the centre of the sphere.
There are five completely symmetric integration rules for the sphere, based on the five regular polyhedra or ‘Platonic Solids’, namely the Regular Tetrahedron, the Regular Hexahedron (cube), the Regular Octahedron, the Regular Dodecahedron and the Regular Icosahedron. In each case the integration rule has the same number of points as there are faces, and we place a microphone capsule at the centre of each face of the polyhedron. This requires 4, 6, 8, 12 and 20 microphone capsules respectively for the five regular polyhedra mentioned. In these symmetrical cases, the weights of the integration rule are all equal, which somewhat simplifies the design of the combining network required to synthesise a particular spherical harmonic.
In such a polyhedral arrangement, the polyhedron may exist physically, or it may be just a conceptual tool to describe the positions of capsules that are suspended in free air, or that are embedded in the surface of a sphere, to give just three examples.
Blumlein's technique for increasing the order of a response can be exemplified by considering two identical omnidirectional capsules separated by a small distance, their outputs being connected to an electrical differencing network. It can be seen that a sound arriving from a direction at right angles to the line joining the two capsules will produce identical outputs from each, and the output of the differencing network will be zero. A sound arriving from along that line will reach one capsule before the other, and the differencing network will thus give a non-zero output on account of the resulting phase difference. Thus a figure-of-eight directional response (or an approximation thereto) is obtained. However at low frequencies, such that the wavelength is long compared with the separation between the capsules, the phase difference will be small and the output of the differencing network will also be small. Blumlein's invention therefore provides for an equaliser to apply bass boost at, ideally, 6 dB/8 ve in order to give a flat frequency response at the final output.
The same principle applies to a spherical, polyhedral or any other arrangement of microphone elements: if the required order of spherical harmonic output is larger than the order provided naturally by the capsules, bass boost is required at 6 dB/8 ve each time the order is increased by one. In particular, to obtain a second order output from zeroth order capsules will require 12 dB/8 ve boost, as described in Rafaely, B., “Design of a Second-Order Soundfield Microphone”, Audio Eng. Soc. 118th Convention (Barcelona 2005), AES preprint #6405, although it is of doubtful practicality if a frequency range spanning several octaves is required.
In the ‘Soundfield’ microphone, the commercial embodiment of the microphone disclosed in GB1512514, large amounts of bass boost are not needed because the required outputs were first order and the individual capsules are also first order (cardioid or sub-cardioid). Nevertheless, equalisation is required at higher frequencies, as is apparent from FIG. 2 of Gerzon, M. A., “The Design of Precisely Coincident Microphone Arrays for Stereo and Surround Sound”, Preprint L-20, 50th convention of the Audio Engineering Society (February 1975).
A symmetrical arrangement of capsules is strongly preferred partly because of simplicity of equalisation. It is possible to use an essentially random array of capsules on the surface of a sphere, or even in its volume (as shown in Laborie, A; Bruno, R; Montoya, S, “A New Comprehensive Approach of Surround Sound Recording” Audio Eng. Soc. 114th Convention, February 2003, AES preprint #5717) an d then to solve linear equations in order to determine the correct (complex) weighting factors to apply to each capsule output. However, in principle, these equations need to be solved separately for each required spherical harmonic output and for each frequency, thus requiring a large number of separately-specified equalisers. The symmetrical approach allows, for each required spherical harmonic output, the capsule outputs to be combined in a frequency independent manner, and then an overall equalisation to be applied that is the same for all harmonics of a given order. In some cases, tractable and implementable expressions can be derived for the equalisation, which is virtually impossible in the random case.
Another advantage of a symmetrical arrangement of capsules relates to spatial (directional) aliasing. When a real sound field is expanded into spherical harmonics, the expansion does not stop at a particular order. The microphone wishes to extract specified lower-order harmonics with minimal contamination from other harmonics, especially from harmonics of an order just slightly higher than that the desired harmonic. For example a dodecahedral array can extract an uncontaminated first order harmonic in the presence of other harmonics of order up to four. There are 1+3+5+7+9 =25 harmonics of order 4, and with a random array it would in general be necessary to use at least 25 capsules in order to reject the 24 unwanted harmonics. A dodecahedral array can do this with just 12 capsules.
Heretofore, it has seemed obvious that if first-order, i.e. directional, capsules are used in a symmetrical 3-D arrangement, then each capsule should have its axis of symmetry (and of maximum sensitivity) pointing outwards radially from the centre, for example as shown diagrammatically in FIG. 1. This arrangement does however have a potential disadvantage, that of producing an acoustic cavity, as will now be explained.
Most practical microphone capsules have a drum-like or disc-like shape. In FIG. 1 the capsules are shown well separated for clarity, but in practice it would be desired to move them closer to the centre of the array in order to maintain the directional performance of the array up to the highest audio frequencies. Making the capsules smaller incurs a penalty in signal-to-noise-ratio, so for capsules of a given size the gap between adjacent capsules will become smaller as they are pulled in, perhaps to the point where adjacent capsules touch. This creates an enclosed air space between the capsules, with access to the outside through the relatively small gaps between the capsules. The mass of the air in the gaps will then resonate with the compliance of the enclosed air, creating a Helmholtz resonance near the top of the audio frequency range. The resonance can in principle be equalised, but it is hard to ensure that there will not be residual inaccuracies in the equalisation, leading to audible coloration.
It might be thought that the resonance could be avoided if the enclosed space were filled with solid material of, for example, spherical or polyhedral shape as discussed earlier. This is an attractive solution if pressure sensors are used, but such an acoustic obstruction will modify the air velocity in its vicinity so as to reduce or nullify the velocity sensitivity of first-order sensors, thus worsening the signal-to-noise ratio at low frequencies.
What is needed is a symmetrical arrangement of first-order sensors that avoids the problems noted above.